r paths
and perpetual deflection therefrom towards any place, is a sure
indication of the existence of some force which from all quarters impels
those bodies towards that place.

That there are centripetal forces actually directed to the bodies of
the sun, of the earth, and other planets, I thus infer.

The moon revolves about our earth, and by radii drawn to its centre
describes areas nearly proportional to the times in which they are
described, as is evident from its velocity compared with its apparent
diameter; for its motion is slower when its diameter is less (and
therefore its distance greater), and its motion is swifter when its
diameter is greater.

The revolutions of the satellites of Jupiter about the planet are more
regular; for they describe circles concentric with Jupiter by equable
motions, as exactly as our senses can distinguish.

And so the satellites of Saturn are revolved about this planet with
motions nearly circular and equable, scarcely disturbed by any
eccentricity hitherto observed.

That Venus and Mercury are revolved about the sun is demonstrable from
their moon-like appearances. And Venus, with a motion almost uniform,
describes an orb nearly circular and concentric with the sun. But
Mercury, with a more eccentric motion, makes remarkable approaches to
the sun and goes off again by turns; but it is always swifter as it is
near to the sun, and therefore by a radius drawn to the sun still
describes areas proportional to the times.

Lastly, that the earth describes about the sun, or the sun about the
earth, by a radius from one to the other, areas exactly proportional to
the times is demonstrable from the apparent diameter of the sun compared
with its apparent motion.

These are astronomical experiments; from which it follows that there are
centripetal forces actually directed to the centres of the earth, of
Jupiter, of Saturn, and of the sun.[2]

That these forces decrease in the duplicate proportion of the distances
from the centre of every planet appears by Cor. vi., Prop. iv., Book
I.[3] for the periodic times of the satellites of Jupiter are one to
another in the sesquiplicate proportion of their distances from the
centre of this planet. Cassini assures us that the same proportion is
observed in the circum-Saturnal planets. In the circum-solar planets
Mercury and Venus, the same proportional holds with great accuracy.

That Mars is revolved about the sun is demonstrated from the p